Answer
See below.
Work Step by Step
The Intermediate Value Theorem says that if a continuous (polynomials are always continuous) function on an interval [a,b] takes values $f(a)$ and $f(b)$ at the endpoints, then the function takes all values between $f(a)$ and $f(b)$ at some point of the interval.
Evaluate the function at the endpoints.
$f(1)=4\cdot1^3-2\cdot1^2-7=4-2-7=-5.$
$f(2)=4\cdot2^3-2\cdot2^2-7=32-8-7=17.$
Since $-5\lt0\lt17$, according to the Intermediate Value Theorem, there must be a $0$ in the given interval.